## Talk by Dr. Jarek Kwapisz (Mathematical Sciences, MSU)

### 09/28/2020  4:10-5:15pm WebEX Talk

Abstract:

Rational functions of the Riemann sphere are quotients of two complex polynomials.  Remarkably, one can often uniquely define such a  function in terms of {\it soft data} given by  its homotopy class relative a finite set of points. What homotopy classes work is a subject of two hallmark results separated by nearly 50 years and due to Bill Thurston and his son Dylan Thurston. The older is the famous {\it Thurston's obstruction},  heralding flight to infinity in the Teichm\"{u}ller space. The newer result, published this summer in Annals of Mathematics, gives a relatively more elementary positive certificate via maps of graphs with rubber band edges, which generate a {\it surface skeleton} of the rational function  when allowed to pull tight.

I will attempt to give an informal description of this rich subject, offering a fascinating mix of  topology, geometry, and analysis.  This talk is a prequel to Lukas' talk (and also loosely connects with my own study of conformal dimension via $p$-resistance).